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Random Variables And Discrete
Probability Distributions
Statistics for Management
and Economics
Chapter 7
Objectives




Random Variables and Probability
Distributions
Bivariate Distributions
Binomial Distribution
Poisson Distribution
Random Variables

A random variable is a function or rule that
assigns a number to each outcome of an
experiment.

Alternatively, the value of a random variable
is a numerical event.

Instead of talking about the coin flipping
event as

{heads, tails} think of it as “the number of
heads when flipping a coin” so we have
{1, 0}
Random Variables



Discrete
one that takes on a
countable number
of values
Values on the roll
of dice: 2, 3, 4, …,
12
Integers



Continuous
one whose values
are not discrete,
not countable
Values resulting
from the event time
taken to walk to
campus (anywhere
from 5 to 30
minutes)
Real Numbers
Probability Distributions

A probability distribution is a table,
formula, or graph that describes the values
of a random variable and the probability
associated with these values.

Since we’re describing a random variable
(which can be discrete or continuous) we
have two types of probability distributions:

Discrete Probability Distribution, (this chapter)

Continuous Probability Distribution (Chapter 8)
Notation

An upper-case letter will represent the name
of the random variable, usually X.

Its lower-case counterpart will represent the
value of the random variable.

The probability that the random variable X
will equal x is: P(X = x)
…or more simply P(x)
Discrete Probability Distributions
The probabilities of the values of a discrete
random variable may be derived by means
of probability tools such as tree diagrams or
by applying one of the definitions of
probability, so long as these two conditions
apply:
Discrete Distributions
A survey of Amazon.com shoppers reveals the following
probability distribution of he number of books purchased per hit:
# of books
x
P(x)
0
0
0.35
1
1
0.25
2
2
0.20
3
3
0.08
4
4
0.06
5
5
0.03
6
6
0.02
7
7
0.01
e.g. P(X=4)
= P(4)
= 0.06
= 6%
What is the probability that
a shopper buys at least one
book, but no more than 3
books?
P(1 ≤ X ≤ 3) = P(1) + P(2) + P(3)
= 0.25 + 0.20 + 0.08
= 0.53
Developing a Probability
Distribution


Probability calculation techniques can be used to
develop probability distributions, for example, a new
game in Vegas is developed where a fair coin is
tossed three times.
What is the probability distribution of the number
of heads if I play this game in Vegas?
Let H denote success, i.e. flipping a head P(H)=.50
Thus HC is not flipping a head, and P(HC)=.50
Probability Distribution

The discrete probability distribution
represents a population

Since we have populations, we can
describe them by computing various
parameters.

E.g. the population mean and
population variance.
Population Mean
Univariate Discrete Probability Distribution

The population mean is the weighted
average of all of its values. The weights are
the probabilities.

This parameter is also called the expected
value of X and is represented by E(X).
Population Variance
Univariate Discrete Probability Distribution

The population variance is calculated similarly. It is
the weighted average of the squared deviations
from the mean.

As before, there is a “short-cut” formulation…

The standard deviation is the same as before:
Population Mean, Variance,
and Standard Deviation
Find the mean, variance, and standard
deviation for the population of the
number of books purchased on
Amazon.com
= 0(0.35) + 1(0.25) + … + 7(0.01)
= 1.51
# of
books
x
P(x)
0
0
0.35
1
1
0.25
2
2
0.20
3
3
0.08
4
4
0.06
5
5
0.03
6
6
0.02
7
7
0.01
= 2.53 = 1.59
= (0-1.51)2*0.35 + (1-1.51)2*0.25 +…+ (7-1.51)2*0.01 = 2.53
Laws of Expected Value
1.
E(c) = c
The expected value of a constant (c) is just the
value of the constant.
2.
E(X + c) = E(X) + c
3.
E(cX) = cE(X)
We can “pull” a constant out of the expected value
expression (either as part of a sum with a
random variable X or as a coefficient of random
variable X).
Laws of Variance
1.
V(c) = 0
The variance of a constant (c) is zero.
2.
V(X + c) = V(X)
The variance of a random variable and a constant is
just the variance of the random variable (per 1
above).
3.
V(cX) = c2V(X)
The variance of a random variable and a constant
coefficient is the coefficient squared times the
variance of the random variable.
Bivariate Distributions

Up to now, we have looked at univariate
distributions, i.e. probability distributions in one
variable.

As you might guess, bivariate distributions are
probabilities of combinations of two variables.

Bivariate probability distributions are also called
joint probability. A joint probability distribution of X
and Y is a table or formula that lists the joint
probabilities for all pairs of values x and y, and is
denoted P(x,y).
P(x,y) = P(X=x and Y=y)
Discrete Bivariate Distribution
As you might expect, the requirements for a
bivariate distribution are similar to a univariate
distribution, with only minor changes to the
notation:
for all pairs (x,y).
Bivariate Distribution
After analyzing several months of sales data, the
owner of an appliance store produced the following
joint probability distribution of the number of
refrigerators and stoves sold daily:
Refrigerators (x)
Stoves (y)
0
1
2
0
0.08
0.14
0.12
0.34
1
0.09
0.17
0.13
0.39
2
0.05
0.18
0.04
0.27
0.22
0.49
0.29
1.00
We interpret these
joint probabilities as
before.
E.g. the probability
that the store sells 0
Refrigerators and 1
Stove in the day is
P(0, 1) = 0.09
Describing the
Bivariate Distribution
We can describe the mean, variance, and standard
deviation of each variable in a bivariate distribution by
working with the marginal probabilities…
x
0
1
2
P(x)
0.22
0.49
0.29
E(x) = 1.07
V(x) = 0.51
x = 0.71
same formulae as for
univariate distributions…
y
0
P(y)
0.34
1
2
0.39
0.27
E(y) = 0.93
V(y) = 0.61
y = 0.78
Covariance
Bivariate Distribution
The covariance of two discrete variables
is defined as:
or alternatively using this shortcut
method:
Coefficient of Correlation
Bivariate Distribution

The coefficient of correlation is
calculated in the same way as
described earlier
Covariance and Correlation
Bivariate Distribution
Compute the covariance and the coefficient of correlation
between the numbers of refrigerators and stoves sold.
COV(X,Y) = (0 –1.07)(0 – 0.93)(0.08) + (1 – 1.07)(0 – 0.93)(0.14) + …
… + (2 – 1.07)(2 – 0.93)(.04) = –0.045
= –0.045 ÷ [(.71)(.78)] = –0.081
There is a weak, negative relationship between the two variables.
Sum of Two Variables
The bivariate distribution allows us to develop the probability
distribution of any combination of the two variables, of particular
interest is the sum of two variables.
If we consider our example of refrigerators and stoves, we can
create a probability distribution…
x+y
0
1
2
3
4
P(x+y)
0.08
0.23
0.34
0.31
0.04
…to answer questions like “what is the probability that two
appliances are sold”?
P(X+Y=2) = P(0,2) + P(1,1) + P(2,0) = 0.05 + 0.17 + 0.12 = 0.34
Laws: Bivariate Distribution
We can derive laws of expected value and
variance for the sum of two variables as
follows…
1.
E(X + Y) = E(X) + E(Y)
2.
V(X + Y) = V(X) + V(Y) + 2COV(X, Y)
If X and Y are independent,
COV(X, Y) = 0 and
thus: V(X + Y) = V(X) + V(Y)
Binomial Distribution
The binomial distribution is the probability
distribution that results from doing a “binomial
experiment”. Binomial experiments have the
following properties:
1.
2.
3.
4.
Fixed number of trials, represented as n.
Each trial has two possible outcomes, a “success”
and a “failure”.
P(success)=p (and thus: P(failure)=1–p), for all
trials.
The trials are independent, which means that the
outcome of one trial does not affect the outcomes of
any other trials.
“Success” and “Failure”

…are just labels for a binomial experiment, there is
no value judgment implied.

For an experiment to b binomial, it simply must
have two possible outcomes: something happens
or something doesn’t happen.

For example a coin flip will result in either heads or
tails. If we define “heads” as success then
necessarily “tails” is considered a failure (inasmuch
as we attempting to have the coin lands heads up).

Other binomial experiment notions:

An election candidate wins or loses

An employee is male or female
Binomial Random Variable

The random variable of a binomial experiment is
defined as the number of successes in the n trials,
and is called the binomial random variable.

E.g. flip a fair coin 10 times…


1) Fixed number of trials  n=10

2) Each trial has two possible outcomes  {heads
(success), tails (failure)}

3) P(success)= 0.50; P(failure)=1–0.50 = 0.50 

4) The trials are independent  (i.e. the outcome of heads
on the first flip will have no impact on subsequent coin
flips).
Hence flipping a coin ten times is a binomial
experiment since all conditions were met.
Binomial Random Variable

The binomial random variable counts the number
of successes in n trials of the binomial experiment.
It can take on values from 0, 1, 2, …, n. Thus, its a
discrete random variable.

To calculate the probability associated with each
value we use combintorics:
for x=0, 1, 2, …, n
Rule of Combinations
The number of combiniations of selecting
X objects out of n objects is given by
n!
X!(n-X)!
Where
[n-factorial] n! = n(n-1)(n-2)…(1) and 0!=1
Rule of Combinations
Often denoted by the symbol
n
X
Thus… with n=4 and x=3…
n!
4!
4x3x2x1
4
=
=
=
X!(n-X)! 3!(4-3)! (3x2x1)(1)
So, there are 4 such sequences, each with the same probability of 0.0009. The
probability of obtaining exactly three tagged order forms is equal to (number of
possible sequences) X (probability of a particular sequence)
Binomial Distribution
n!
P(X) =
X!(n-X)!
pX(1-p)n-X
Where…
P(X) = probability of X successes given the parameters n and p
n = sample size
p = probability of success
(1-p) = probability of failure
X = number of successes in the sample (X = 0, 1, 2, …, n)
E.C.K. Pharmaceutical Company
If the likelihood of a tagged order form is 0.1, what is the
probability that three tagged order forms are found in the
sample of four orders?
P(3) =
4!
3!(4-3)!
p3(1-p)4-3 = 0.0036
If the likelihood of a tagged order form is 0.1, what is the
probability that three or more (i.e., at least three) tagged
order forms are selected out of the sample of four order
forms? Notation? How do we find this?
Cumulative Probability

Thus far, we have been using the binomial
probability distribution to find probabilities for
individual values of x. To answer the question:
“Find the probability that the order is tagged”

requires a cumulative probability, that is, P(X ≤ x)

What is the probability that we find fewer than three
tagged order forms in the sample of four orders?

Thus, we want to know what is: P(X ≤ 3) to
answer
Cumulative Probability
P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
Use the binomial probability to find…
4!
P(0) =
p0(1-p)4-0 = 0.6561
0!(4-0)!
P(1) =
P(2) =
4!
1!(4-1)!
4!
2!(4-2)!
p1(1-p)4-1 = 0.2916
p2(1-p)4-2 = 0.0486
P(X ≤ 3) = 0.9963
Is there another way to get this?
Binomial Table




The probabilities listed in the tables are
sometimes cumulative, i.e. P(X ≤ x) – x
(sometimes k) is the row index (our book is
like this).
In some cases, the probabilities are not
cumulative – P(X=k) is shown.
This should be indicated in a heading on the
table.
Take out your packet of tables now.
Binomial Table
For a binomial table that gives cumulative probabilities
for P(X ≤ k)…
P(X = k) = P(X ≤ k) – P(X ≤ [k–1])
Likewise, for probabilities given as P(X ≥ k), we have:
P(X ≥ k) = 1 – P(X ≤ [k–1])
However, if the table does not give cumulative
probabilities, as we saw in the example, in order to
find P(X ≤ k) you have to add the k-1, etc
probabilities.
=BINOMDIST()
Excel Function…
There is a binomial distribution function in Excel that can
also be used to calculate these probabilities. For example:
What is the probability that two or fewer orders are tagged?
# successes
# trials
P(success)
cumulative
(i.e. P(X≤x)?)
P(X≤2)=.9963
Binomial Distribution
As you might expect, statisticians have developed
general formulas for the mean, variance, and
standard deviation of a binomial random variable.
They are:
Poisson Distribution
Named for Simeon Poisson, the Poisson distribution
is a discrete probability distribution and refers to the
number of events (a.k.a. successes) within a
specific time period or region of space. For
example:
o
o
o
The number of cars arriving at a service station in 1 hour.
(The interval of time is 1 hour.)
The number of flaws in a bolt of cloth. (The specific region
is a bolt of cloth.)
The number of accidents in 1 day on a particular stretch
of highway. (The interval is defined by both time, 1 day,
and space, the particular stretch of highway.)
The Poisson Experiment
Like a binomial experiment, a Poisson experiment
has four defining characteristic properties:
1.
2.
3.
4.
The number of successes that occur in any interval is
independent of the number of successes that occur in any
other interval.
The probability of a success in an interval is the same for
all equal-size intervals
The probability of a success is proportional to the size of
the interval.
The probability of more than one success in an interval
approaches 0 as the interval becomes smaller.
Poisson Distribution
The Poisson random variable is the number of
successes that occur in a period of time or an
interval of space in a Poisson experiment.
successes
E.g. On average, 96 trucks arrive at a border
crossing every hour.
time period
E.g. The number of typographic errors in a new
textbook edition averages 1.5 per 100 pages.
successes (?!)
interval
Poisson Probability Distribution
The probability that a Poisson random
variable assumes a value of x is given by:
and e is the natural logarithm base.
FYI:
=Poisson Excel Function